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The zero section proves injectivity of the Thom unit orbit

Statement

Assume AC. For the stable Thom class U∈M⁰, the map ν:A→M, a↦aU, is injective in every degree.

Facts & Assumptions

Given: AC; a nonzero homogeneous $a \in A^{d}$ expanded in the admissible basis; the integer $r = d + 1$ ; the space $X = ({R P}^{L})^{r}$ with $L \geq d + 1$ ; the external sum $E = ⨁_{i = 1}^{r} {p r}_{i}^{*} γ_{1}$ ; and the based zero section $z_{+} : X_{+} \to T (E)$ .

[F1]

The admissible terms of $a$ have distinct leading monomials on $P = x_{1} \dots x_{r}$ , so $a (P) \neq 0$ (Admissible square actions have distinct leading monomials, Admissible composites present the mod-two square algebra).

[F2]

The stable Grassmannian classification supplies a classifying map $g$ of $E$ with $g^{*} γ_{r} ≅ E$ , and the pullback Thom map pulls the universal Thom class back to the Thom class of $E$ ; the Euler class is the zero-section pullback of the Thom class, the top Stiefel–Whitney class computes the mod-two Euler class, and the Whitney formula gives $w_{r} (E) = P$ (Real and complex vector bundles are classified by stable Grassmannians, Euler class by zero-section pullback of the Thom class, The mod-two Euler class is the top Stiefel–Whitney class, Whitney sum formula for Stiefel–Whitney classes, Tautological degree-one class on a real projective bundle, Stiefel–Whitney classes from the projective-bundle relation, Naturality of Stiefel–Whitney classes, Naturality and uniqueness of Thom classes, The Thom prespectrum of the universal real and oriented bundles).

[F3]

Squares are natural for pullbacks of Thom classes and act on the inverse limit componentwise (Degreewise mod-two cohomology of the universal real Thom prespectrum, Stable Steenrod squares on universal Thom cohomology); AC underlies the choices of models (The Axiom of Choice).

Proof

technique · direct

1.1givenF1

Take a nonzero homogeneous a∈A^d. Set r=d+1 and X=(RP^L)^r with L≥d+1. Let x_i be the degree-one generator from factor i and P=x₁⋯x_r. By the leading-monomial lemma, the admissible terms in a have distinct leading monomials on P, so a(P)≠0.

2.1step 1.1F2F3

Let E=⊕{i=1}^r pr_i^*γ₁ over X. The stable Grassmannian classification gives a classifying map g:X→BO(r) with g^*γ_r≅E. The prespectrum setup’s pullback Thom map T(E)→T_r pulls u_r back to u_E. The based zero section z+:X_+→T(E) pulls the normalized Thom class back to the mod-two Euler class: z_+^u_E=e_r(E)=w_r(E)=x₁⋯x_r=P. The first equality is the published Euler definition by zero-section pullback; the second is the published mod-two Euler/top-Stiefel–Whitney theorem; For each line L_i=pr_i^γ₁, P(L_i)=X and its tautological line is L_i. The rank-one projective-bundle relation is x_(L_i)+w₁(L_i)=0, hence w₁(L_i)=x_(L_i). The factor map X→RP^L→RP^∞ classifies L_i, so the tautological-class definition gives x_(L_i)=x_i. The rank convention gives w_j(L_i)=0 for j>1. The Whitney formula now gives w(E)=∏(1+x_i), whose top-degree part is P. These are precisely the rank-one Stiefel–Whitney definition and the tautological-class definition; no identification of w₁ is inferred from the tautological definition alone. Naturality of every Sq composite, hence of a, now gives z_+^ T(g)^ (a u_r)=a(P)≠0.

3.1step 2.1F3∎

Thus a u_r≠0. If aU were zero in M, every inverse-limit component would vanish, in particular its rank-r component a u_r; contradiction. So ν is injective in each degree and hence on the graded direct sum. For an inhomogeneous element, its distinct degree components remain distinct in M and cannot cancel.

Depends on

Used by

Stable unoriented Thom cohomology is free over the square algebra Theorem

Dependency tree · two levels

87 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Tom Weston, An Introduction to Cobordism Theory (standard reference, not scraped)
John Milnor and James Stasheff, Characteristic Classes (standard reference, not scraped)